I installed a fibreglass sink on top of a vanity and now I need to attach the 2 pieces of backsplash (also fibreglass), one on each side of the countertop. What sort of adhesive do I use and do I use adhesive where the backsplash touches the sidewall or just where it touches the countertop?

Use a good tile mastic on the wall that you are going to put the fiberglass on. Use a trowel with small notches in it. After the fiberglass has time to dry, (maybe two days) then use a tub and tile caulk and make a seam where the top and the back splash meet.